Snap all vertices to grid when translating (not center of selection)
You can move the Origin of the object, which is currently your "center of object", to the grid.
In Edit Mode, select a vertex on the grid, then press Shift S and Snap to Selected.
Then in Object Mode, go to the Tools Tab of the Tools Editor, Set Origin, and Origin to Cursor.
I'm in 2.79 - hope you can find the commands in 2.8.